🤖 AI Summary
This work addresses discrete combinatorial optimization problems subject to linear constraints—including inequalities—by proposing the constrained Matrix Product State (cMPS), a probabilistic modeling framework that exactly embeds the feasible domain into a sparse tensor structure. Methodologically, it introduces the novel concept of “quantum regions” to uniformly encode arbitrary linear constraints and establishes a new canonical form enabling efficient fusion/decomposition of tensor blocks and optimal truncation. The proposed unsupervised constrained optimization training framework achieves model lightweighting, strong robustness against overfitting, and accelerated convergence. Empirically, on the quadratic knapsack problem, cMPS substantially outperforms state-of-the-art nonlinear integer programming solvers. Furthermore, comprehensive experiments validate its scalability and generalization capability across complex combinatorial optimization tasks.
📝 Abstract
In this study, we introduce a novel family of tensor networks, termed extit{constrained matrix product states} (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.